Timeline for How many geometric structures on manifolds are there?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2015 at 22:58 | answer | added | Mike Cocos | timeline score: 1 | |
| Jan 30, 2015 at 12:17 | comment | added | Robert Bryant | The OP's question was about $G$-structures, not geometric structures. As, the OP states "A large number of [geometric structures] are captured by the notion of a $G$-structure", so the OP makes the distinction. Also, one should be careful; the 'classification' of geometric structures, à la Thurston's 8 geometries in dimension $3$, makes assumptions (such as, for example, that the geometric structure can be found on some compact manifold) that often are not made explicit in superficial expositions. Without these assumptions, there are far more than $8$ transitive pseudo-groups in dimension $3$. | |
| Jan 30, 2015 at 8:52 | comment | added | Qiaochu Yuan | @Ryan: I think the term "geometric structure" is a little misleading here. In particular I don't think the OP means a geometric structure in the sense of Thurston, but means e.g. an orientation or a spin structure. | |
| Jan 30, 2015 at 6:56 | comment | added | Ryan Budney | @Ben McKay: I did not realize people were interested in geometric structures that are not locally homogeneous. | |
| Jan 30, 2015 at 6:51 | comment | added | Ben McKay | @RyanBudney: Your comment assumes that the geometric structures are locally homogeneous. There is no classification of $G$-structures on 4-manifolds. | |
| Jan 30, 2015 at 6:30 | comment | added | Ryan Budney | Geometric structures on 4-manifolds were classified some time ago. It was a ?Cambridge? or perhaps ?Oxford? student that did not remain in mathematics. Jon Hillman wrote up the results in his book "4-manifolds, geometries and knots" maths.ed.ac.uk/~aar/papers/hillman.pdf The upshot is geometric 4-manifolds are much like geometric 3-manifolds, in that they are "generalized Seifert fibre spaces over surfaces" with a few small exceptions, like hyperbolic manifolds. | |
| Jan 30, 2015 at 0:59 | comment | added | Robert Bryant | @user41626: On a compact $4$-manifold, you can reduce to an $\mathrm{SO}(3)\times 1$ structure if and only if $M$ is orientable and has vanishing Euler characteristic. To reduce to $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SO}(2)$, you need to know that the manifold is orientable and possesses a continuous rank-$2$ subbundle $E\subset TM$ (and conversely). For example, $S^4$ does not possess a $G$-structure of either kind. In fact, the only connected $G\subset\mathrm{GL}(4,\mathbb{R})$ that work in this case must, up to conjugacy, contain an $\mathrm{SO}(4)$. | |
| Jan 30, 2015 at 0:19 | answer | added | Qiaochu Yuan | timeline score: 8 | |
| Jan 29, 2015 at 23:25 | comment | added | Mariano Suárez-Álvarez | You should probably make explicit what makes your question different from «what are the coverings of linear Lie groups?» which is what the comments are converging to. | |
| Jan 29, 2015 at 23:24 | comment | added | Paul Reynolds | Although of course, torsion-freeness is required for the holonomy to be $G$. Note also that any distribution (in the sense of Frobenius) is a structure, and these are studied in many contexts. | |
| Jan 29, 2015 at 23:12 | comment | added | Alex Degtyarev | Also, I think that any reduction of $SO$ to a subgroup is basically a question about the holonomy group of the metric, and I think that such things are studied. | |
| Jan 29, 2015 at 23:10 | comment | added | Alex Degtyarev | Well, some reductions of $SO(2n+1)$ to $SO(2n)$ are called contact structures and are indeed studied a lot. Although I do agree that not everything is studied/worth studying, but this does not seem to be relevant to your question title :) | |
| Jan 29, 2015 at 22:14 | review | Close votes | |||
| Jan 30, 2015 at 15:30 | |||||
| Jan 29, 2015 at 22:12 | comment | added | Jon Middleton | I agree. However I don't imagine that every possible structure group reduction has been studied, nor do I imagine that they all are worthy of study. If $M$ is a $4$-manifold, then I can reduce its frame bundle to, say, $SO(3)\times 1$, or $SL(2,\mathbb{R})\times SO(2)$. Has anyone looked at these $G$-structures? | |
| Jan 29, 2015 at 21:57 | comment | added | Alex Degtyarev | According to your definition, a list of $G$-structures is a list of subgroups of $GL(n)$. (In fact, one should also include at least various coverings of these subgroups, to accommodate for all sorts of $\mathrm{Spin}$-structures.) IMHO, a very exotic but popular structure is the $G_2$-structure on $7$-manifolds. | |
| Jan 29, 2015 at 21:45 | history | asked | Jon Middleton | CC BY-SA 3.0 |